What is the difference between on and off-policy deterministic actor-critic?
The twist here is that the $a_{t+1}$ in (11) and the $\mu(s_{t+1})$ in (16) are the same and actually the $a_t$ in the on-policy case and the $a_t$ in the off-policy case are different.

The key to the understanding is that in on-policy algorithms you have to use actions (and generally speaking trajectories) generated by the policy in the updating steps (to improve the policy itself). This means that in the on-policy case $a_i = \mu(s_{i})$ (in equations 11-13).

Whereas in the off-policy case you can use any trajectory to improve your value/action-value functions, which means that the actions $a_t$ can be generated by any distribution $a_t~\pi(s_t,a_t)$. In (16) the algorithm explicitly states however that the action-value function ($Q^w$) has to be evaluated at $\mu_{s_{t+1}}$ (just like in the on-policy case) and not at $a_t$ which was the actual action in the trajectory generated by policy $\pi$.